Κυριακή, 4 Δεκεμβρίου 2011

Constructible sets

Πηγή : Scribd Badiou-Alain-Politics-A-nonexpressive-dialectics

There is a very clear mathematical example of this relation between desire and law, between different forms of existence, which is really interesting. Don’t be afraid, it’s very simple. I think mathematics is very often something which is linked to ter- ror. And I am always speaking from a non-terrorist conception of mathematics…

Suppose that we are in the theory of sets – we have a theory of the pure multiplicity – and suppose we consider one set, no matter what set; a multiplicity absolutely ordinary. The interesting thing is that with some technical means we can for- malise the idea of a subset of this set which has a clear name. So the question of the relation between existence and clear name has a possible formalisation in the field of the mathematical theory of sets. Precisely, to have a clear name is to be defined by one clear formula. It was an invention of the greatest logician of the last century, Kurt Gödel. He named that sort of subset a con- structible subset; a constructible subset of a set is a set which has a clear description. And generally speaking we name con- structible set a set which is a constructible subset of another set.

So, we have here the possibility of what I name a great law. What is a great law? A great law is a aw of laws, if you want: the law of what is really the possibility of a law. And we have a sort of math- ematical example of what is that sort of law, which is not only a law of things or subjects, but a law for laws. The great law takes the form of an axiom, the name of which is the axiom of con- structibility and which is very simple. This axiom is: all sets are constructible. You know that is a decision about existence: you decide that exist only sets that are constructible, and you have as a simple formula a simple decision about existence. All sets are constructible, that is the law of laws. And this is really a possibil- ity. You can decide that all sets are constructible. Why

Because all mathematical theorems which can be demonstrated in the general theory of sets can also be demonstrated in the field of con- structible sets. So all that is true of sets in general is in fact true for only constructible sets. So – and it’s very interesting about the question, the general question of the law – we can decide that all sets are constructible, or if you like that all multiplicity is under the law, and we lose nothing: all that is true in general is true with the restriction to constructible sets. If you lose nothing, if the field of truth is the same under the axiom of constructibility, we can say something like: the law is not a restriction of life and thinking; under the law, the liberty of living and thinking is the same. And the mathematical model of that is that we don’t lose anything when we have the affirmation that all sets are constructible, that is to say all parts of sets are constructible, that is to say finally all parts have a clear definition. And as we have a general classifica- tion of parts, a rational classification of parts; classification of society if you want – without any loss of truth. At this point there is a very interesting fact, a pure fact. Practically no mathematician admits the axiom of constructibili- ty. It’s a beautiful order, in fact, it’s a beautiful world: all is con- structible. But this beautiful order does not stimulate the desire of a mathematician, as conservative as he might be. And why? Because the desire of the mathematician is to go beyond the clear order of nomination and constructibility. The desire of the mathematician is also the desire for a mathematical monster. They want a law, certainly – difficult to do mathematics without laws – they want a law but the desire to find some new mathematical monster is beyond this law.

The mathematical example is very striking. After Gödel, the def- inition of constructible sets, and the refusal of the axiom of con- structibility by the majority of mathematicians, the question of the mathematician’s desire becomes: how can I find a non-con- structible set? And you see the difficulty, which is of great politi- cal consequence. The difficulty is, how can we find some mathe- matical object without clear description of it, without name, with- out place in the classification: how to find an object the character- istic of which is to have no name, to not be constructible, and so on. And the very complex and elegant solution was found in the sixties by Paul Cohen. He found an elegant solution to name, to identify, a set which is not constructible, which has no name, which has no place in the great classification of predicates, a set which is without specific predicate. It was a great victory of desire against law, in the field of law itself, the mathematical field. And like many things, many victories of this type, it was in the sixties. And Paul Cohen gives the nonconstructible sets a very beautiful name: generic sets. And the invention of generic sets is something in the revolutionary actions of the sixties.

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